Estimates of norms of log-concave random matrices with dependent entries
Marta Strzelecka
TL;DR
The paper addresses the problem of estimating the expected operator norm $\mathbb{E}\|X\|_{p',q}$ of random matrices with dependent entries, where $X_{ij}=A_{ij}Y_{ij}$ and the rows $Y_i$ are i.i.d. isotropic log-concave vectors. It extends Gaussian-based results to this log-concave setting by leveraging a Guédon–Rudelson–Mendelson–Pajor–Tomczak-Jaegermann framework with $E=\ell_{p'}^n$, and by controlling auxiliary quantities through moment-regularity and Sudakov-type bounds. The central contribution is an explicit bound $\mathbb{E}\|X\|_{p',q} \le C(p,q)\big[ (\log m)^{1/q}\max_i\|A_i\|_p + \max_j\|A^{(j)}\|_q + (\log m)^{1+1/q}\mathbb{E}\max_{i,j}|X_{ij}|\big]$, optimal up to logarithmic factors, with further results for Gaussian mixtures and for the unconditional-entries regime. These results yield dimension-dependent norms for a broad class of random matrices beyond independent Gaussian entries, with potential applications in high-dimensional probability and random matrix theory.
Abstract
We prove estimates for $\mathbb{E} \| X: \ell_{p'}^n \to \ell_q^m\|$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d. isotropic log-concave rows. This generalises the result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide the analogue bound for $m\times n$ random matrices, which entries form an unconditional vector in $\mathbb{R}^{mn}$. We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.